In three spatial dimensions, particles are classified into bosons and fermions. Bosons have integer spin and obey Bose-Einstein statistics, while fermions have half-integer spin and obey Fermi-Dirac statistics. In two-dimensional systems, particles with properties continuously interpolating from bosons to fermions are theoretically allowed to exist and they are called anyons. Anyons are characterized by a fractional spin, or more generally, by fractional quantum numbers. Apart from the fundamental interest, the main motivation for studying anyons comes from their potential importance in fault-tolerant topological quantum computing. The work presented in this thesis contributes to the study of less traditional schemes for realization and manipulation of anyons. The first part focuses on new mechanisms for the realization and signatures of anyons in non-interacting systems. We propose an experimental realization of the original Wilczek’s model for Abelian anyons. This proposal is implemented in two-dimensional electron gas placed in a perpendicular magnetic field which gives rise to the integer quantum Hall effect. Then we present exact solutions of a model for synthetic anyons in a non-interacting quantum many-body system. The model is represented by the Hamiltonian for non-interacting electrons in two dimensions, in a uniform magnetic field, pierced with specially tailored localized probes, solenoids with a magnetic flux that is a fraction of the flux quantum. Here we show that synthetic anyons cannot be considered as emergent quasiparticles. The second part of the thesis concentrates on a system of one-dimensional bosons coupled to synthetic gauge fields. In particular, we investigate a system of strongly interacting bosons placed on a one-dimensional ring pierced by a synthetic magnetic flux tube. An external localized delta-function potential barrier is placed on the ring and we explore the Berry phase associated to its adiabatic motion. The barrier produces a cusp in the density, where we show that the corresponding missing charge cannot be identified as a quasihole. This result is associated with the previous studies of synthetic anyons in non-interacting systems.